Question: $ -2.\overline{19} \div 0.\overline{38} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 100x &= -219.192...\\ x &= -2.192...\end{align*} $ $\begin{align*} 99x &= -217 \\ x &= -\dfrac{217}{99}\end{align*} $ $\begin{align*} 100y &= 38.3838...\\ y &= 0.3838...\end{align*} $ $\begin{align*} 99y &= 38 \\ y &= \dfrac{38}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{217}{99} \div \dfrac{38}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{217}{99} \times \dfrac{99}{38} = {?} $ $ \phantom{-\dfrac{217}{99} \times \dfrac{38}{99}} = \dfrac{-217 \times 99}{99 \times 38} $ $ \phantom{-\dfrac{217}{99} \times \dfrac{38}{99}} = \dfrac{-217 \times \cancel{99}} {\cancel{99} \times 38} $ $ \phantom{-\dfrac{217}{99} \times \dfrac{38}{99}} = -\dfrac{217}{38} $